Questions tagged [arithmetic-geometry]
Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.
2,211 questions
1 vote
0 answers
124 views
A p-adic analogue of Tate's conjecture
Let $\mathcal{A}\rightarrow S$ be a family of abelian varieties over $\overline{\mathbb{F}}_p$ with constant Newton polygon $\xi$, say with slope zero part dimension $n$. Then one can consider the $p$-...
- 103
3 votes
0 answers
302 views
Are discrete topological modules (light) solid?
Recall two definitons: Let $\mathrm P_{\mathbb Z} = \mathbb Z[\overline{\mathbb N}] / \mathbb Z[\infty]$ denote the free condensed abelian group on null sequences. A condensed abelian group $M$ is ...
- 71
5 votes
0 answers
185 views
Locally Separated Morphism of Sheaves
Definition 9.2.2 of Scholze and Weinstein's Berkeley Lectures on $p$-adic Geometry says "Consider the site Perf of perfectoid spaces of characteristic $p$ with the pro-ètale topology. A map $f:\...
3 votes
0 answers
112 views
Kernel of $\operatorname{Br}(F) \to \prod_v \operatorname{Br}(F_v)$ and the Tate–Shafarevich group
Let $X$ be a smooth projective curve over a number field $K$ which has a $K$-rational point, and let $F = K(X)$ be its function field. Consider the map $$ \operatorname{Br}(F) \longrightarrow \prod_v \...
- 161
0 votes
0 answers
128 views
Tate/semisimplicity conjecture for resolution of nodal quartic surface
Thanks to work of many people (Madapusi, Charles, Maulik...) the Tate conjecture is now known for K3 surfaces over any finitely generated field. A smooth quartic surface is K3. Consider instead a ...
- 1,106
4 votes
1 answer
179 views
Are abelian varieties over $\mathbb{F}_q[t]$ which have the same $L$-function isogenous?
Faltings' isogeny theorem implies that two abelian varieties over a number field $K$ which have the same characteristic polynomial of Frobenius at almost all primes are isogenous. If $K = \mathbb{Q}$ ...
- 1,106
2 votes
0 answers
139 views
Find a ring R such that Spec R is homeomorphic to Spa(Z,Z)
I'm following Scholze-Weinstein's Berkeley notes (https://www.math.uni-bonn.de/people/scholze/Berkeley.pdf). And there is a theorem by Huber (Thm 2.3.3) in the notes that says the adic spectrum $\...
- 21
8 votes
0 answers
242 views
Is the solvable closure of $\mathbb{F}_p(t)$ PAC?
It is a famous open problem in field arithmetic whether $\mathbb{Q}^{\mathrm{solv}}$, the solvable closure of $\mathbb{Q}$, is pseudo algebraically closed (PAC). That is, whether every absolutely ...
- 161
2 votes
0 answers
227 views
Why is the notion of $(\phi,\Gamma)$-module useful?
I have a, hopefully not so stupid, feeling that the notion of $(\phi,\Gamma)$-modules is an abstract model for $p$-adic Galois representations. I am wondering what is good for us when we work with ...
- 1,496
2 votes
0 answers
90 views
Quasi-isogeny corresponding to the image of the Frobenius in the Dieudonné module of a $p$-divisible group
Let $X$ be a $p$-divisible group over a perfect field $k$ of characteristic $p>0$. Let $M(X)$ be the covariant Dieudonné module of $X$. Thus, $M(X)$ is a finite free $W(k)$-module equipped with a $\...
- 799
3 votes
0 answers
257 views
Need copy of old preprint: "Arithmetic of Fermat varieties I, Fermat motives and p-adic cohomologies" by Suwa-Yui
Illusie's article in volume II of the Grothendieck Festschrift cites a 1988 preprint: "Arithmetic of Fermat varieties I, Fermat motives and p-adic cohomologies", by Suwa-Yui. It does not ...
- 1,106
3 votes
1 answer
261 views
Chebotarev density for function fields
In On $l$-independence of Algebraic Monodromy Groups in Compatible Systems of Representations, they stated such an example (Example 6.3): Let $X$ be a connected normal scheme of finite type over $\...
- 103
3 votes
0 answers
164 views
The quadratic forms corresponding to canonical (Neron-Tate) height of an elliptic curve
Surely this is well-known in the arithmetic geometry community, which I (unfortunately) do not regard as a member of, so I will include some basic exposition for the laymen (including myself). We deal ...
- 29.9k
2 votes
0 answers
191 views
Explicit description of the crystal associated to the universal object on a Lubin-Tate space
Let $X:=\mathrm{LT}_h$ be the Lubin-Tate space of the (unique) formal group law of height $h$ over $k:=\overline{\mathbb{F}}_p$. The adic generic fiber $X_{\eta}=\mathrm{LT}_{h,\eta}$ is then a rigid-...
- 29
0 votes
0 answers
113 views
Confusion on theorem in paper on monodromy of $p$-rank strata of moduli of curves
Application 5.7 in "Monodromy of the $p$-rank strata of the moduli space of curves", by Achter and Pries, seems to imply that for $g \ge 3$, there exists a supersingular smooth projective ...
- 1,106
1 vote
0 answers
254 views
Vanishing of Weil–Châtelet group over solvably closed field
Let $A$ be a connected algebraic group defined over a solvably closed field $K$. Are there any results concerning the vanishing of the WC-group $H^1(G_K, A)$?
- 161
4 votes
1 answer
234 views
If two non-isomorphic cubic surfaces over a field $k$ have all their lines defined over $k$, can they become isomorphic over the algebraic closure?
If two non-isomorphic smooth cubic surfaces over a field $k$ each contain 27 lines defined over $k$, can they become isomorphic over the algebraic closure?
- 1,106
4 votes
0 answers
255 views
$L$-function form of Tate conjecture for divisors on abelian varieties
Let $k$ be the function field of a $d$-dimensional regular integral finite-type scheme $Y$ over $\mathbb{Z}$. Conjecture 2 in Tate's paper in the Woods Hole proceedings predicts (among other things) ...
- 1,106
1 vote
0 answers
240 views
Does $\text{Gal}(K(F[\pi^{\infty}])/K \cong \text{Gal}(K(G[\pi^{\infty}])/K$ imply $F[\pi^n] \cong G[\pi^n]$?
Let $K$ be a $p$-adic number field with ring of integers $\mathcal{O}_K$ and uniformizer $\pi$. Let $F$ and $G$ be two $d$-dimensional formal groups of height $h$ over $\mathcal{O}_K$. Denote: \begin{...
- 429
4 votes
1 answer
433 views
Semisimplicity of crystalline Frobenius
Let $X$ denote a smooth proper scheme over a finite field $k$ of characteristic $p$. It is my understanding that it is not generally known if the Frobenius action on the rationalized crystalline ...
- 3,346
1 vote
0 answers
63 views
Horizontal sections of the unit‑root sub‑isocrystal for GM-connection, and equivalence with local systems
Let $Y\to X$ denote a relative abelian scheme, with $X$ smooth proper over $k$ a finite field of characteristic $p$. Let $\mathcal{V} := R^1_{\mathrm{cr}}f_*\textbf{1}[1/p]$ denote the Gauss-Manin $F$-...
- 3,346
1 vote
0 answers
205 views
Nilpotence of De Rham cohomology in positive characteristic
I am trying to show that if $\pi: X\rightarrow S, f: S\rightarrow T$ are smooth morphisms over positive characteristic. Then $(\mathbb{H}^i_{dR}(X/S), \nabla_{GM})$ is nilpotent, in the sense of Katz (...
- 103
7 votes
1 answer
326 views
How big can the degree of the field of definition of a morphism of abelian variety be?
Excuse me if some parts of the questions are well known, I am still a PhD student and have a lot to learn. This question came to me while doing research on the Coleman's conjecture (as presented in a ...
2 votes
1 answer
322 views
Canonical descent of Serre-Tate canonical lift
Deligne seems to say on page 239 here that the Serre-Tate canonical lift of an ordinary abelian variety over the algebraic closure $\overline{k}$ of a finite field of characteristic $p$ canonically ...
- 1,106
5 votes
0 answers
132 views
what is space moduli space of higher-dimensional formal group law?
Let $k$ be a perfect field and $W(k)$ be the ring of Witt vector and define the ring of formal power series $$R:=W(k)[[v_1, \cdots, v_{h-1}]]$$ in $h-1$ variables. Then there is a canonical surjection ...
- 429
2 votes
0 answers
194 views
What is the Dirichlet density of the reducible locus in the moduli of polarized abelian varieties in positive characteristic?
By section 2 of https://swc-math.github.io/aws/2024/2024KaremakerNotes.pdf, for $q = p^r$ a prime power, any $g, d \ge 1$, and any $n \ge 3$ prime to $q$, there exists a stratified fine moduli space $...
- 571
5 votes
1 answer
222 views
$\mathbb{Z}_p^{\mathbb{N}}$-extension and formal Drinfeld module
Thanks for your help. I'm reading the paper Iwasawa main conjecture for the Carlitz cyclotomic extension and applications (https://arxiv.org/pdf/1412.5957). I find three questions about the motivation ...
- 473
5 votes
1 answer
403 views
When does the Kodaira symbol determine the Tamagawa number?
Let $E/K$ be an elliptic curve over a local field. I understand that the Kodaira type of $E/K$ refers to the isomorphism class of the special fiber of the Néron minimal model of $E/K$ as a scheme over ...
- 53
1 vote
0 answers
169 views
Rational points of abelian varieties over finite fields
Let $A$ be an abelian variety over a finite field $k$ and $\ell$ a prime number. Fix a natural number $n\in \mathbb{N}$ and denote by $k_{n}$ the field $k(A[\ell^{n}](\overline k))$. Let $Q\in A[\ell^{...
- 327
5 votes
2 answers
481 views
Do Frobenius elements at closed points topologically generate the fundamental group?
For $X$ a regular (or perhaps just normal) variety over Spec $\mathbb{Z}$, is the etale fundamental group of $X$ topologically generated by Frobenius elements at closed points? This is true when $X$ ...
- 571
2 votes
0 answers
254 views
Why is the category of Weil-étale sheaves abelian with enough injectives?
When I study Weil-étale cohomology put forward by S. Lichtenbaum in the paper The Weil-étale topology on schemes over finite fields, I find two questions. Let $X$ be a scheme of finite type over a ...
- 473
2 votes
0 answers
157 views
Deligne local constant
Let $F$ be a non-Archimedean local field. For a multiplicative character $\chi$ of $F^\times$, let $a(\chi)$ be the conductor of $\chi$. For a nontrivial additive character $\phi$ of $F$, we denote ...
- 53
1 vote
0 answers
97 views
Line bundle and height on abelian variety
Let $A$ be an abelian variety over a number field $K$ of dimension $g$. Consider $\mathcal{L}$ be a symmetric ample line on $A$ that gives an embedding $ \ i_{\mathcal{L}} : A(\overline{K}) \...
- 275
2 votes
0 answers
189 views
How could we get the Weil group for global function fields?
Thanks for your help. I know the definition of Weil group and how to get it for $p$-adic local field cases. By reading the answer https://math.stackexchange.com/a/2898449/1007843, I write down my ...
- 473
7 votes
0 answers
447 views
Greenberg schemes of mixed characteristic varieties
Let $k$ be a perfect field of characteristic $p>0$, and $R=W(k)$ the ring of $p$-typical Witt vectors. For any $R$-scheme $X$ and $k$-algebra $A$, set $$\operatorname{Gr}_n(X)(A)=\operatorname{Hom}...
- 315
2 votes
0 answers
130 views
Section of abelian variety over local field
Let $K$ be a non-archimedean local field (e.g., $\mathbb{Q}_p$ ), $A / K$ an abelian variety of dimension $g$. Consider $\mathcal{L}$ a line bundle on $A$ and $F \in \Gamma(A, \mathcal{L})$ a global ...
- 275
3 votes
0 answers
223 views
Neron model of dual abelian variety
Let $A/K$ be an abelian variety where $K = \operatorname{Frac}(R)$ for some discrete valuation ring $R$. Let $\mathcal{A}$ be its Neron model. Assume that $A$ has potential good reduction, so that the ...
- 393
2 votes
0 answers
155 views
Infinitude of points on a rank 1 elliptic curve satisfying a "geometric" condition
A friend recently nerd-sniped me with a (seemingly elementary) geometry question: Let $\triangle ABC$ (with corresponding side lengths $a$, $b$, and $c$) be an obtuse triangle with $\angle C > 90^\...
- 43
4 votes
0 answers
286 views
Hecke action on cohomology
This question is regarding a reduction step in Proposition 5.2.6 in [Scholze] (that is V.2.6 in the arxiv version). Notation: Let's restrict to the case where $F=\mathbb{Q}$ for simplicity Let $X_K$ ...
- 939
3 votes
1 answer
439 views
Suggestion for book/lecture notes
I am a number theory graduate student and I've got interest in Diophantine Geometry recently. I don't have any background in Algebraic Geometry and due to which I am struggling a lot. I want to learn ...
- 275
1 vote
0 answers
79 views
Shioda-Inose structure in characteristic 2
I am looking for some work on the Shioda-Inose structure in characteristic $2$. In particular, I am interested in if there is some analogue of Kummer sandwich theorem in characteristic $2$, at least ...
- 191
2 votes
0 answers
164 views
Lifting automorphic forms on Shimura subvarieties to automorphic forms on the Shimura variety
Forgive my ignorance, I am new to the subject. I have a need for determining the existence of certain lifts of elliptic modular forms to Hilbert modular forms (over a real quadratic field). The ...
- 91
3 votes
2 answers
216 views
Representing elements of Tate-Shafarevich group of $E$ as translates of $E$ in abelian variety
Let $E$ be an elliptic curve over a number field $K$ (I’m more generally interested in global fields). Cremona and Mazur in section 3 of these notes explain that any element $C$ of the Tate-...
- 1,106
1 vote
0 answers
120 views
Height of a projective point evaluated at a homogeneous polynomial
Let $F$ be a homogeneous polynomial of degree $L$ in $K[X_0, X_1,...., X_N]$ and $P$ is a point of $N$ dimensional projective space over $K$. If the logarithm Weil height of $P$ is $h(P)$ and $h(F)$ ...
- 275
5 votes
0 answers
283 views
$n$-torsion points of ample divisors on abelian varieties over $\overline{\mathbb{F}_{p}}$
Let $A$ be an abelian variety over $\overline{\mathbb{F}_{p}}$ of dimension $d$ and $D\subseteq A$ an effective divisor on $A$ such that $D$ is not a union of translates of abelian subvarieties. Let $...
- 327
4 votes
1 answer
265 views
algebraic fundamental group of Raynaud generic fiber
Let $k$ be a perfect field of characteristic $p$. Let $X$ be a quasi-projective smooth variety over the Witt ring $W=W(k)$ ($K=\mathrm{Frac}(W)$). Let $\mathcal X$ be the $p$-adic formal completion of ...
- 471
5 votes
0 answers
303 views
Gabber’s $\ell'$-alteration theorem and varieties over function fields acquiring regular models after base change
I have seen it claimed that one can use Gabber’s $\ell'$-alteration theorem (theorem X.2.4 here) to prove the following statement: if $X$ is a smooth projective variety over a global function field $K$...
- 1,106
2 votes
0 answers
169 views
Is the torsion points $F[p^{\infty}]$ of a formal group law Zariski dense in $\mathfrak{m}_{\mathbb{C}_p}^d$?
Let $F$ be a $d$-dimensional formal group of finite height over the ring of $p$-adic integers $\mathbb{Z}_p$. Let $\mathfrak{m}_{\mathbb{C}_p}$ be the unit disk in $\mathbb{C}_p$. Since $F$ the formal ...
- 429
2 votes
2 answers
517 views
Idempotents in Chow ring
In his introductory paper on motives, James Milne gives a "first attempt" on the construction of the category of motives and he writes that in this attempt, images on idempotents are ...
- 721
7 votes
1 answer
484 views
Group cohomology for finite étale group schemes
Let $k$ be a field. Let $G,M$ be finite étale group schemes over $\mathrm{Spec}(k)$, in other words finite groups equipped with a continuous action of the absolute Galois group $\Gamma_k$ of $k$. ...
- 1,068